Simplify cube root of 54x^4y^6- cube root of 24x^3y^9+ cube root of 128x^4y^6

$\sqrt[3]{54 x^{4} y^{6}} - \sqrt[3]{24 x^{3} y^{9}} + \sqrt[3]{128 x^{4} y^{6}}$

Step 1

Simplify each term.

Rewrite $54 x^{4} y^{6}$ as ${(3 x y^{2})}^{3} \cdot (2 x)$ .

Factor $27$ out of $54$ .

$\sqrt[3]{27 (2) x^{4} y^{6}} - \sqrt[3]{24 x^{3} y^{9}} + \sqrt[3]{128 x^{4} y^{6}}$

Rewrite $27$ as $3^{3}$ .

$\sqrt[3]{3^{3} \cdot 2 x^{4} y^{6}} - \sqrt[3]{24 x^{3} y^{9}} + \sqrt[3]{128 x^{4} y^{6}}$

Factor out $x^{3}$ .

$\sqrt[3]{3^{3} \cdot 2 (x^{3} x) y^{6}} - \sqrt[3]{24 x^{3} y^{9}} + \sqrt[3]{128 x^{4} y^{6}}$

Rewrite $y^{6}$ as ${(y^{2})}^{3}$ .

$\sqrt[3]{3^{3} \cdot 2 (x^{3} x) {(y^{2})}^{3}} - \sqrt[3]{24 x^{3} y^{9}} + \sqrt[3]{128 x^{4} y^{6}}$

Move $x$ .

$\sqrt[3]{3^{3} \cdot 2 (x^{3}) {(y^{2})}^{3} x} - \sqrt[3]{24 x^{3} y^{9}} + \sqrt[3]{128 x^{4} y^{6}}$

Move $2$ .

$\sqrt[3]{3^{3} (x^{3}) {(y^{2})}^{3} \cdot 2 x} - \sqrt[3]{24 x^{3} y^{9}} + \sqrt[3]{128 x^{4} y^{6}}$

Rewrite $(3^{3} (x^{3})) {(y^{2})}^{3}$ as ${(3 x y^{2})}^{3}$ .

$\sqrt[3]{{(3 x y^{2})}^{3} \cdot 2 x} - \sqrt[3]{24 x^{3} y^{9}} + \sqrt[3]{128 x^{4} y^{6}}$

Add parentheses.

$\sqrt[3]{{(3 x y^{2})}^{3} \cdot (2 x)} - \sqrt[3]{24 x^{3} y^{9}} + \sqrt[3]{128 x^{4} y^{6}}$

Pull terms out from under the radical.

$3 x y^{2} \sqrt[3]{2 x} - \sqrt[3]{24 x^{3} y^{9}} + \sqrt[3]{128 x^{4} y^{6}}$

Rewrite $24 x^{3} y^{9}$ as ${(2 x y^{3})}^{3} \cdot 3$ .

Factor $8$ out of $24$ .

$3 x y^{2} \sqrt[3]{2 x} - \sqrt[3]{8 (3) x^{3} y^{9}} + \sqrt[3]{128 x^{4} y^{6}}$

Rewrite $8$ as $2^{3}$ .

$3 x y^{2} \sqrt[3]{2 x} - \sqrt[3]{2^{3} \cdot 3 x^{3} y^{9}} + \sqrt[3]{128 x^{4} y^{6}}$

Rewrite $y^{9}$ as ${(y^{3})}^{3}$ .

$3 x y^{2} \sqrt[3]{2 x} - \sqrt[3]{2^{3} \cdot 3 x^{3} {(y^{3})}^{3}} + \sqrt[3]{128 x^{4} y^{6}}$

Move $3$ .

$3 x y^{2} \sqrt[3]{2 x} - \sqrt[3]{2^{3} x^{3} {(y^{3})}^{3} \cdot 3} + \sqrt[3]{128 x^{4} y^{6}}$

Rewrite $2^{3} x^{3} {(y^{3})}^{3}$ as ${(2 x y^{3})}^{3}$ .

$3 x y^{2} \sqrt[3]{2 x} - \sqrt[3]{{(2 x y^{3})}^{3} \cdot 3} + \sqrt[3]{128 x^{4} y^{6}}$

Pull terms out from under the radical.

$3 x y^{2} \sqrt[3]{2 x} - (2 x y^{3} \sqrt[3]{3}) + \sqrt[3]{128 x^{4} y^{6}}$

Multiply $2$ by $- 1$ .

$3 x y^{2} \sqrt[3]{2 x} - 2 (x y^{3} \sqrt[3]{3}) + \sqrt[3]{128 x^{4} y^{6}}$

Rewrite $128 x^{4} y^{6}$ as ${(4 x y^{2})}^{3} \cdot (2 x)$ .

Factor $64$ out of $128$ .

$3 x y^{2} \sqrt[3]{2 x} - 2 x y^{3} \sqrt[3]{3} + \sqrt[3]{64 (2) x^{4} y^{6}}$

Rewrite $64$ as $4^{3}$ .

$3 x y^{2} \sqrt[3]{2 x} - 2 x y^{3} \sqrt[3]{3} + \sqrt[3]{4^{3} \cdot 2 x^{4} y^{6}}$

Factor out $x^{3}$ .

$3 x y^{2} \sqrt[3]{2 x} - 2 x y^{3} \sqrt[3]{3} + \sqrt[3]{4^{3} \cdot 2 (x^{3} x) y^{6}}$

Rewrite $y^{6}$ as ${(y^{2})}^{3}$ .

$3 x y^{2} \sqrt[3]{2 x} - 2 x y^{3} \sqrt[3]{3} + \sqrt[3]{4^{3} \cdot 2 (x^{3} x) {(y^{2})}^{3}}$

Move $x$ .

$3 x y^{2} \sqrt[3]{2 x} - 2 x y^{3} \sqrt[3]{3} + \sqrt[3]{4^{3} \cdot 2 (x^{3}) {(y^{2})}^{3} x}$

Move $2$ .

$3 x y^{2} \sqrt[3]{2 x} - 2 x y^{3} \sqrt[3]{3} + \sqrt[3]{(4^{3} (x^{3})) {(y^{2})}^{3} \cdot 2 x}$

Rewrite $(4^{3} (x^{3})) {(y^{2})}^{3}$ as ${(4 x y^{2})}^{3}$ .

$3 x y^{2} \sqrt[3]{2 x} - 2 x y^{3} \sqrt[3]{3} + \sqrt[3]{{(4 x y^{2})}^{3} \cdot 2 x}$

Add parentheses.

$3 x y^{2} \sqrt[3]{2 x} - 2 x y^{3} \sqrt[3]{3} + \sqrt[3]{{(4 x y^{2})}^{3} \cdot (2 x)}$

Pull terms out from under the radical.

$3 x y^{2} \sqrt[3]{2 x} - 2 x y^{3} \sqrt[3]{3} + 4 x y^{2} \sqrt[3]{2 x}$

Step 2

Add $3 x y^{2} \sqrt[3]{2 x}$ and $4 x y^{2} \sqrt[3]{2 x}$ .

$7 x y^{2} \sqrt[3]{2 x} - 2 x y^{3} \sqrt[3]{3}$

Simplify cube root of 54x^4y^6- cube root of 24x^3y^9+ cube root of 128x^4y^6

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